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Conjecture: a(2*n-1) = 0 for all n > 0. Also, a(2*n) is always a fourth power, , and a(2*n) is nonzero when n > 9.

Zhi-Wei Sun also made could prove the following general conjecturerelated result:

(i) Let m be any positive even integer, and let D(m, n) denote the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod m or not. Then (-1)^{n*(n-1)/2}*D(m,n) is always an m-th power. (It is easy to see that D(m,n) = 0 if m does not divide n^2.)

(ii) For any integers m > 0 and r, the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a prime congruent to r mod m or not, is a square times (-1)^{n*(n-1)/2}.

Cf. A002144, A069191, A228591, A228552, A228557, A228559, A228561, A228615, A228616.

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Cf. A002144, A228550, A069191, A228552, A228557, A228559, A228561.

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(i) Let m be any positive even integer, and let D(m, n) denote the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod m or not. Then (-1)^{m/2*n*(n-1)/2}*D(m,n) is always an m-th power. (It is easy to see that D(m,n) = 0 if m does not divide n^2.)

(ii) For any integers m > 0 and r, the absolute value of the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a prime congruent to r mod m or not, is always a square times (-1)^{n*(n-1)/2}.

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(i) Let m be any positive even integer, and let D(m, n) denote the n X n determinant with (i,j)-entry equal to 1 or 0 according as i + j is a prime congruent to 1 mod m or not. Then |(-1)^{m/2*n*(n-1)/2}*D(m, n)| is always an m-th power, and moreover D(m,n) is nonnegative if 4 | m. (It is easy to see that D(m,n) = 0 if m does not divide n^2.)

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